26 Art. L— T. Takenouchi : 



Hence evidently these h invariants are all relatively prime not 

 only to one another but also to p, while the r invariants of 21 are 

 all powers of p. Now, let A u A a> ■■■, A s be a system of bases of 

 9Ji=2I33 and let their orders be a 1} « 2 , •■•, a s respectively. Then, 

 from the constitution of the invariants, it follows that any two a' s 

 are certainly relatively prime to each other unless both of them 

 contain powers of p. But since there are only r powers of p in 

 the invariants, if we suppose s>r, at least two of a's, say a, and a.,, 

 must be relatively prime to each other. Then we may replace 

 the two bases A l and A 2 by a single one A^A % of order a,a 2 , and 

 thus the number s can be diminished by unity. On the other 

 hand, however, s cannot be made less than r, for no one of a's 

 can contain more than one of the r powers of p at the same time. 

 We conclude therefore that the least possible number of bases of 

 Wl is equal to r. Hence, in order that m=p" may admit of primi- 

 tive roots, the necessary and sufficient condition is r=l. 



Therefore, from (44), it follows that primitive roots exist only 

 in the following cases: 



(i) fd=l, n>d + k, provided p + œ p-1 = (mod. p'*" 1 " 1 ) has no solution, 



(ii) fd=\, n = d + k, 



(iii) f(n- [— ]]=1, l<n<:d+k, 



(iv) n = l. 



(iv) shews that every prime ideal has primitive roots. This is 

 the well-known theorem, which we have already made use of. 

 From (i), (ii), (iii), we get/=l. Therefore, 



unless p be a prime ideal of the first degree, there exists no primitive 



root of p n when «>1. 



Putting /= 1 in (i), we get d=l, and consequently 



k= — -y =1. Hence primitive roots may exist when d=l, 



n>2, provided p + x p ~ ] =0 (mod, p d+1 ) has no solution. Now, as 

 we have remarked in the last paragraph, in order that this con- 

 gruence may have a solution, it is necessary that d = (mod. p — 1); 

 whence follows p=2, since here d=l. That this condition p = 2 



